|
Order of Magnitude Calculations:
This topic is one of my favorites, because it is so applicable. Every scientist I know uses order-of-magnitude (OOM) calculations when they are brainstorming or discussing topics. They help provide a quick sense of scale, a check on feasibility, and a sense of the important parameters in considering magnitude of the final result.
Goal: Find a reasonable, ballpark estimate without delving into too much complexity. Should be do-able on the back of an envelope (maybe with the help of a calculator). The end result should be within a factor of 10 of the "true" answer.
Rule 1: Keep track of the units that you are using. These will often guide your attempt at finding a solution. Consider the units of the final answer that you are seeking and the units of "known" or approximate-able variables that are related. These units obey the same rules of algebra that variables obey. Formally, the algebraic manipulation of units is called "dimensional analysis."
Rule 2: Learn the scale of some basic properties. The important properties to know will vary from field to field. Below is a list of the ones that I think are important.
|
Population |
World |
6.5 billion people |
| USA |
300 million people |
|
Rates |
Walking |
1 m/s |
| Driving |
20 m/s |
| Tectonic plates |
1-10 cm/yr |
| Mountain uplift |
0.1-1 mm/yr |
|
Densities |
Air |
1 kg/m³ |
| Water |
1000 kg/m³ |
| Bedrock |
2700 kg/m³ |
|
Distances |
E-W width of USA |
3000 miles, 4800 km |
| N-S height of USA |
1250 miles, 2000 km |
| Earth's radius |
6370 km |
|
Time |
Age of Earth |
4.5 billion years |
| Last Glacial Maximum |
18,000 years ago |
| Last interglacial period |
125,000 years ago |
| Direct environmental observations |
~100 years |
| USA election cycles |
2-6 years |
|
Conversions |
Distance |
3.28 ft/m |
| 5280 ft/mile |
| Mass |
2.2 lbs/kg |
Rule 3: Make simplifying assumptions to make the problem solvable.
Example 1: How long before a hybrid car becomes economical?
(Extra cost)-(tax incentives)=(gallons saved/year)*(cost/gallon)*(years)
Rearranging this equation, we find
(years)=[(Extra cost)-(tax incentives)]/[(gallons saved/year)*(cost/gallon)]
We can plug in numbers for a car like the Honda Civic hybrid:
hybrid: cost = $22,000; fuel efficiency = 50 miles/gallon
non-hybrid: cost = $17,000; fuel efficiency = 35 miles/gallon
tax credit: $2100
gasoline costs: $3/gallon
average miles driven: 12,000 miles/year
The conclusion is that you would save $300/year and it would take you ~16 years to make up the price difference from buying a hybrid instead of a non-hybrid vehicle. You should be able to see in the equation that the number of years goes down if gas prices go up, you save more gallons of gasoline each year (through driving more each year or improving your fuel efficiency more), if the price difference between hybrids and non-hybrids comes down, and/or the tax incentives improve.
Example 2: How much fuel would be saved by everyone in the USA using 100 gallons of gasoline less each year?
(gallons saved)=(gallons saved/car)*(USA population)*(fraction that owns cars)
There is about one car for every two Americans, or a fraction of 0.5. Given a US population of 300 million and 100 gallons/year of savings, 15 billion gallons/year would be conserved. This is approximately 1 billion barrels of oil (20 gallons of gasoline from each barrel of oil). Given that ANWR petroleum reserves are estimated at 16 billion barrels, this is a significant amount of oil.
Example 3: What was the annual per capita rate of earth moving in building the Roman road network?
(mass of earth moved)=(width of roads)*(length of roads)*(thickness of the road)*(density of paving material)
(rate of earth moving)=(mass of earth moved)/[(time it took to build the roads)*(# of people in Roman civilization)]
The actual values can be found by doing some archaeological research. Keep in mind that this was the most advanced civilization of its time, and therefore is not representative of the global rate of earth moving during that time period. Also keep in mind that this is just one of the ways that Romans moved stone and sediment.
|
